35 Line Bundles on Cpn

35 Line Bundles on CPn

n n Recall CP = {[z0 : z1 : ... : zn]}. Here CP , the set of all complex lines passing through the origin, is deﬁned to a quotient space Cn+1 −{0}/ ∼ where

n+1 (z0, z1, ..., zn), (z0, z1, ..., zn) ∈ C −{0}

z , z , ..., z λ z , z , ..., z λ ∈ C −{ } are equivalent if and only if ( 0 1 ne)=e ( 0 e 1 n) with 0 . 1 Holomorphic line bundle of degree k over CP The holomorphic line bundle OCP1 (k) 1 e e e 1 of degree k ∈ Z over CP is constructed explicitly as follows. Let CP = U0 ∪ U1 be the standard open covering. Using the standard atlas, take two “trivial” pieces U0 × C and 0 1 i U1 × C with coordinates (z0,ζ ) and (z1,ζ ) where zi is the base coordinate of Ui and ζ is ∗ 1 the ﬁbre coordinate over Ui - and “glue them together” over the set C = CP \{0, ∞} by the identiﬁcation 1 1 0 ζ z0 = , ζ = k . (121) z1 (z1)

This line bundle is usually denoted OCP1 (k), or simply O(k) for brevity. The total space k k H of O(k) is the identiﬁcation space; thus H consists of the disjoint union U0 U1 of 2 0 1 two copies of C = C × C, with respective coordinates (z0,ζ ) and (z1,ζ ), which have been ∗ ` glued together along C × C according to (121). The transition function from z0 to z1 is 1 g01(z0)= k ; (z0)

1 the other is g10(z1)= (z1)k .

n CPn In general for higher dimensional case, let {U} = {Uα}α=0 be the standard atlas on . The line bundle OCPn (k) is deﬁned by the transition functions

z k g ([z]) = β . (122) αβ z α

z0 z1 k 1 In particular, when n = 1, g01([z]) = g01([z0 : z1]) = g01([ z1 : 1]) = z0 , i.e., g01(λ) = λk which is the same formula above. The tautological line bundle over projective spaces For each point [Z] in CP1, let 2 1 ℓ[Z] be the line in C represented by this point. Show that the family {ℓ[Z] | [Z] ∈ CP } may be regarded as a holomorphic line bundle in the following manner: Set L = {([Z],ζ) ∈

192 1 2 CP × C | ζ ∈ ℓ[Z]}, with projection to the ﬁrst factor. This holomorphic line bundle is called the tautological bundle. Since there is a natural holomorphic map from L ∩ (CP1 × C) to C, by using the fact that a holomorphic map s : CP1 → C2 must be constant to show that L has no sections.

In general for higher dimensional case, let us deﬁne a line bundle over CPn by assigning a one dimensional vector space at each point [ℓ] ∈ CPn:

{([ℓ], v) ∈ CPn × Cn+1 | v ∈ ℓ}.

Here ℓ is a complex line in Cn+1 passing through the origin. This deﬁnes a line bundle L, which is called the tautological line bundle over CPn. As above, such line bundle L has no section.

CPn n Let us ﬁnd out the transition functions of the line bundle L. Let = ∪α=0Uα be the standard open covering. A canonical trivialization of L over Uα is given by

−1 θα : π (Uα) ≃ Uα × C ([ℓ], z) 7→ ([ℓ], zα)

−1 C C where z = (z0, ..., zn). The transition maps gαβ([ℓ]) = θα ◦ θβ ([ℓ]) : → are given by zα w 7→ · w, where [ℓ] = [z : ... : zn]. Then by (122), it concludes that k = −1 and hence zβ 0 the tautological line bundle L = OCPn (−1).

The dual line bundles When k < 0, the bundle O(k) has no sections. In fact, since the k tautological line bundle is O(−1), then O(k) = O(−1) ⊗| |. As in (2), any holomorphic section of O(−1) induces a holomorphic map from CPn → C so that such map must be constant and hence zero. Similarly, any holomorphic section of O(−1)⊗|k| alsi induces a holomorphic map from CPn to C so that it is a zero section.

When k = 0, O(k) is trivial bundle so that any holomorphic section is a holomorphic map from CPn to C. By the maximum principle, the trivial bundle has zero section only.

d When k > 0, O(k) always has holomorphic sections. Let Symn denote the space of homogeneous polynomials of degree d on Cn. Let f be any homogeneous polynomial of d d z0, ..., zn of degree d. Then f = (zα) fα = (zβ) fβ holds on Uα ∩ Uβ where fα is the non- d homogeneous polynomial obtained by dividing f with (zα) . In other words, fα = gαβfβ zβ d holds on U ∩ U , where g = is the transition function for OCPn (k). α β αβ zα 193 k Therefore, the space of holomorphic sections of OCPn (k) is equal to Symn+1 for k > 0 and is trivial otherwise. In particular, the homogeneneous coordinate functions are sections of the line bundle O(−1). Here we use the notation

O(1) := O(−1)∗,

O(k) := O(1)⊗k = O(1) ⊗ ... ⊗ O(1) (k − times) and O(−k) := O(−1)⊗k.

Basic properties

Proposition 35.1

n + k if k ≥ 0, 0 CPn dim H ( , O(k)) =  n ! 0, ifk< 0.

 Later we’ll prove that O(k) are all holomorphic line bundles over CPn. For general vector bundles over CP1, we have the following famous result.

Theorem 35.2 (Grothendieck lemma) Every holomorphic vector bundle E on CP1 is iso- morphic to a holomorphic vector bundle of the form ⊕O(aj). The ordered sequence a1 ≥ a2 ≥ ... ≥ ar is uniquely determined.

Grothendieck lemma no longer holds for n > 1. A vector bunlde on CPn for n ≥ 2 is a direct sum of line bundle is equivalent to some geometric and topological condition.

194